?

Average Error: 51.03% → 12.38%
Time: 1.1min
Precision: binary64
Cost: 104708

?

\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
↓
\[\begin{array}{l} t_1 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \sqrt[3]{\tan k}\\ t_3 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ t_4 := \frac{t}{t_3}\\ t_5 := \sqrt[3]{t_1 \cdot \sin k} \cdot t_2\\ \mathbf{if}\;t \leq -4.6 \cdot 10^{+26}:\\ \;\;\;\;\frac{\frac{\frac{2}{t_4}}{t_5}}{{\left(\frac{\sqrt[3]{t_1}}{\frac{\frac{t_3}{t}}{t_2 \cdot \sqrt[3]{\sin k}}}\right)}^{2}}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-190}:\\ \;\;\;\;-2 \cdot \left(\frac{\frac{\cos k}{k}}{{\sin k}^{2}} \cdot \left(\ell \cdot \frac{\ell}{k \cdot \left(-t\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_3 \cdot \frac{2}{t}}{t_5}}{{\left(t_4 \cdot t_5\right)}^{2}}\\ \end{array} \]
(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))
↓
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (+ 2.0 (pow (/ k t) 2.0)))
        (t_2 (cbrt (tan k)))
        (t_3 (pow (cbrt l) 2.0))
        (t_4 (/ t t_3))
        (t_5 (* (cbrt (* t_1 (sin k))) t_2)))
   (if (<= t -4.6e+26)
     (/
      (/ (/ 2.0 t_4) t_5)
      (pow (/ (cbrt t_1) (/ (/ t_3 t) (* t_2 (cbrt (sin k))))) 2.0))
     (if (<= t 3.6e-190)
       (* -2.0 (* (/ (/ (cos k) k) (pow (sin k) 2.0)) (* l (/ l (* k (- t))))))
       (/ (/ (* t_3 (/ 2.0 t)) t_5) (pow (* t_4 t_5) 2.0))))))
double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0));
}
↓
double code(double t, double l, double k) {
	double t_1 = 2.0 + pow((k / t), 2.0);
	double t_2 = cbrt(tan(k));
	double t_3 = pow(cbrt(l), 2.0);
	double t_4 = t / t_3;
	double t_5 = cbrt((t_1 * sin(k))) * t_2;
	double tmp;
	if (t <= -4.6e+26) {
		tmp = ((2.0 / t_4) / t_5) / pow((cbrt(t_1) / ((t_3 / t) / (t_2 * cbrt(sin(k))))), 2.0);
	} else if (t <= 3.6e-190) {
		tmp = -2.0 * (((cos(k) / k) / pow(sin(k), 2.0)) * (l * (l / (k * -t))));
	} else {
		tmp = ((t_3 * (2.0 / t)) / t_5) / pow((t_4 * t_5), 2.0);
	}
	return tmp;
}

public static double code(double t, double l, double k) {
	return 2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) + 1.0));
}
↓
public static double code(double t, double l, double k) {
	double t_1 = 2.0 + Math.pow((k / t), 2.0);
	double t_2 = Math.cbrt(Math.tan(k));
	double t_3 = Math.pow(Math.cbrt(l), 2.0);
	double t_4 = t / t_3;
	double t_5 = Math.cbrt((t_1 * Math.sin(k))) * t_2;
	double tmp;
	if (t <= -4.6e+26) {
		tmp = ((2.0 / t_4) / t_5) / Math.pow((Math.cbrt(t_1) / ((t_3 / t) / (t_2 * Math.cbrt(Math.sin(k))))), 2.0);
	} else if (t <= 3.6e-190) {
		tmp = -2.0 * (((Math.cos(k) / k) / Math.pow(Math.sin(k), 2.0)) * (l * (l / (k * -t))));
	} else {
		tmp = ((t_3 * (2.0 / t)) / t_5) / Math.pow((t_4 * t_5), 2.0);
	}
	return tmp;
}

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) + 1.0)))
end
↓
function code(t, l, k)
	t_1 = Float64(2.0 + (Float64(k / t) ^ 2.0))
	t_2 = cbrt(tan(k))
	t_3 = cbrt(l) ^ 2.0
	t_4 = Float64(t / t_3)
	t_5 = Float64(cbrt(Float64(t_1 * sin(k))) * t_2)
	tmp = 0.0
	if (t <= -4.6e+26)
		tmp = Float64(Float64(Float64(2.0 / t_4) / t_5) / (Float64(cbrt(t_1) / Float64(Float64(t_3 / t) / Float64(t_2 * cbrt(sin(k))))) ^ 2.0));
	elseif (t <= 3.6e-190)
		tmp = Float64(-2.0 * Float64(Float64(Float64(cos(k) / k) / (sin(k) ^ 2.0)) * Float64(l * Float64(l / Float64(k * Float64(-t))))));
	else
		tmp = Float64(Float64(Float64(t_3 * Float64(2.0 / t)) / t_5) / (Float64(t_4 * t_5) ^ 2.0));
	end
	return tmp
end

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[t_, l_, k_] := Block[{t$95$1 = N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Tan[k], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(t / t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(N[Power[N[(t$95$1 * N[Sin[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * t$95$2), $MachinePrecision]}, If[LessEqual[t, -4.6e+26], N[(N[(N[(2.0 / t$95$4), $MachinePrecision] / t$95$5), $MachinePrecision] / N[Power[N[(N[Power[t$95$1, 1/3], $MachinePrecision] / N[(N[(t$95$3 / t), $MachinePrecision] / N[(t$95$2 * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.6e-190], N[(-2.0 * N[(N[(N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l * N[(l / N[(k * (-t)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$3 * N[(2.0 / t), $MachinePrecision]), $MachinePrecision] / t$95$5), $MachinePrecision] / N[Power[N[(t$95$4 * t$95$5), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]]]]]

\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
↓
\begin{array}{l}
t_1 := 2 + {\left(\frac{k}{t}\right)}^{2}\\
t_2 := \sqrt[3]{\tan k}\\
t_3 := {\left(\sqrt[3]{\ell}\right)}^{2}\\
t_4 := \frac{t}{t_3}\\
t_5 := \sqrt[3]{t_1 \cdot \sin k} \cdot t_2\\
\mathbf{if}\;t \leq -4.6 \cdot 10^{+26}:\\
\;\;\;\;\frac{\frac{\frac{2}{t_4}}{t_5}}{{\left(\frac{\sqrt[3]{t_1}}{\frac{\frac{t_3}{t}}{t_2 \cdot \sqrt[3]{\sin k}}}\right)}^{2}}\\

\mathbf{elif}\;t \leq 3.6 \cdot 10^{-190}:\\
\;\;\;\;-2 \cdot \left(\frac{\frac{\cos k}{k}}{{\sin k}^{2}} \cdot \left(\ell \cdot \frac{\ell}{k \cdot \left(-t\right)}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{t_3 \cdot \frac{2}{t}}{t_5}}{{\left(t_4 \cdot t_5\right)}^{2}}\\


\end{array}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Split input into 3 regimes

  2. if t < -4.6000000000000001e26

    1. Initial program 36.35

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

    2. Simplified44.04

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
      Proof

      [Start]36.35

      \[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      *-commutative [=>]36.35

      \[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      distribute-rgt1-in [<=]36.35

      \[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      *-commutative [=>]36.35

      \[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]

      associate-*l* [=>]36.35

      \[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

      associate-*l* [=>]43.97

      \[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]

      distribute-lft-in [=>]43.97

      \[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left(\tan k \cdot 1 + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]

      *-rgt-identity [=>]43.97

      \[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\color{blue}{\tan k} + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      distribute-lft-in [=>]43.97

      \[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\sin k \cdot \tan k + \sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

    3. Applied egg-rr21.25

      \[\leadsto \color{blue}{\frac{1}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \cdot \frac{2}{\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

    4. Simplified21.25

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}\right)}^{2}}} \]
      Proof

      [Start]21.25

      \[ \frac{1}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \cdot \frac{2}{\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      associate-*l/ [=>]21.25

      \[ \color{blue}{\frac{1 \cdot \frac{2}{\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}}} \]

    5. Applied egg-rr21.24

      \[\leadsto \frac{\frac{\frac{2}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\color{blue}{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \sqrt[3]{\tan k}}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}\right)}^{2}} \]

    6. Applied egg-rr2.58

      \[\leadsto \frac{\frac{\frac{2}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \sqrt[3]{\tan k}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\left(\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \sqrt[3]{\tan k}\right)}\right)}^{2}} \]

    7. Applied egg-rr2.55

      \[\leadsto \frac{\frac{\frac{2}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \sqrt[3]{\tan k}}}{{\color{blue}{\left(\frac{\sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}}}{\frac{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}}{\sqrt[3]{\sin k} \cdot \sqrt[3]{\tan k}}}\right)}}^{2}} \]

    if -4.6000000000000001e26 < t < 3.60000000000000007e-190

    1. Initial program 78.17

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

    2. Simplified78.98

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
      Proof

      [Start]78.17

      \[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      *-commutative [=>]78.17

      \[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      distribute-rgt1-in [<=]78.17

      \[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      *-commutative [=>]78.17

      \[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]

      associate-*l* [=>]78.13

      \[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

      associate-*l* [=>]78.83

      \[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]

      distribute-lft-in [=>]78.83

      \[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left(\tan k \cdot 1 + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]

      *-rgt-identity [=>]78.83

      \[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\color{blue}{\tan k} + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      distribute-lft-in [=>]78.83

      \[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\sin k \cdot \tan k + \sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

    3. Taylor expanded in t around 0 44.09

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]

    4. Simplified38.46

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\frac{\ell}{\frac{t}{\ell}}}{{\sin k}^{2}}\right)} \]
      Proof

      [Start]44.09

      \[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

      times-frac [=>]46.6

      \[ 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)} \]

      unpow2 [=>]46.6

      \[ 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right) \]

      *-commutative [=>]46.6

      \[ 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{{\ell}^{2}}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

      associate-/r* [=>]45.39

      \[ 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{\sin k}^{2}}}\right) \]

      unpow2 [=>]45.39

      \[ 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\frac{\color{blue}{\ell \cdot \ell}}{t}}{{\sin k}^{2}}\right) \]

      associate-/l* [=>]38.46

      \[ 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\color{blue}{\frac{\ell}{\frac{t}{\ell}}}}{{\sin k}^{2}}\right) \]

    5. Applied egg-rr24.6

      \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\cos k}{k} \cdot \left(-\ell\right)}{{\sin k}^{2} \cdot \left(k \cdot \frac{-t}{\ell}\right)}} \]

    6. Simplified24.2

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\frac{\cos k}{k}}{{\sin k}^{2}} \cdot \left(\frac{-\ell}{-k \cdot t} \cdot \ell\right)\right)} \]
      Proof

      [Start]24.6

      \[ 2 \cdot \frac{\frac{\cos k}{k} \cdot \left(-\ell\right)}{{\sin k}^{2} \cdot \left(k \cdot \frac{-t}{\ell}\right)} \]

      times-frac [=>]26.87

      \[ 2 \cdot \color{blue}{\left(\frac{\frac{\cos k}{k}}{{\sin k}^{2}} \cdot \frac{-\ell}{k \cdot \frac{-t}{\ell}}\right)} \]

      associate-*r/ [=>]24.18

      \[ 2 \cdot \left(\frac{\frac{\cos k}{k}}{{\sin k}^{2}} \cdot \frac{-\ell}{\color{blue}{\frac{k \cdot \left(-t\right)}{\ell}}}\right) \]

      associate-/r/ [=>]24.2

      \[ 2 \cdot \left(\frac{\frac{\cos k}{k}}{{\sin k}^{2}} \cdot \color{blue}{\left(\frac{-\ell}{k \cdot \left(-t\right)} \cdot \ell\right)}\right) \]

      distribute-rgt-neg-out [=>]24.2

      \[ 2 \cdot \left(\frac{\frac{\cos k}{k}}{{\sin k}^{2}} \cdot \left(\frac{-\ell}{\color{blue}{-k \cdot t}} \cdot \ell\right)\right) \]

    if 3.60000000000000007e-190 < t

    1. Initial program 43.37

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

    2. Simplified49.2

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
      Proof

      [Start]43.37

      \[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      *-commutative [=>]43.37

      \[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      distribute-rgt1-in [<=]43.37

      \[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      *-commutative [=>]43.37

      \[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]

      associate-*l* [=>]43.33

      \[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

      associate-*l* [=>]49.03

      \[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]

      distribute-lft-in [=>]49.03

      \[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left(\tan k \cdot 1 + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]

      *-rgt-identity [=>]49.03

      \[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\color{blue}{\tan k} + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      distribute-lft-in [=>]49.03

      \[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\sin k \cdot \tan k + \sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

    3. Applied egg-rr25.82

      \[\leadsto \color{blue}{\frac{1}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \cdot \frac{2}{\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

    4. Simplified25.86

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}\right)}^{2}}} \]
      Proof

      [Start]25.82

      \[ \frac{1}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \cdot \frac{2}{\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      associate-*l/ [=>]25.82

      \[ \color{blue}{\frac{1 \cdot \frac{2}{\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}}} \]

    5. Applied egg-rr25.86

      \[\leadsto \frac{\frac{\frac{2}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\color{blue}{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \sqrt[3]{\tan k}}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}\right)}^{2}} \]

    6. Applied egg-rr11.27

      \[\leadsto \frac{\frac{\frac{2}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \sqrt[3]{\tan k}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\left(\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \sqrt[3]{\tan k}\right)}\right)}^{2}} \]

    7. Applied egg-rr35

      \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(2 \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)} - 1}}{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \sqrt[3]{\tan k}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \sqrt[3]{\tan k}\right)\right)}^{2}} \]

    8. Simplified11.27

      \[\leadsto \frac{\frac{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \frac{2}{t}}}{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \sqrt[3]{\tan k}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \sqrt[3]{\tan k}\right)\right)}^{2}} \]
      Proof

      [Start]35

      \[ \frac{\frac{e^{\mathsf{log1p}\left(2 \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)} - 1}{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \sqrt[3]{\tan k}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \sqrt[3]{\tan k}\right)\right)}^{2}} \]

      expm1-def [=>]11.56

      \[ \frac{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)\right)}}{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \sqrt[3]{\tan k}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \sqrt[3]{\tan k}\right)\right)}^{2}} \]

      expm1-log1p [=>]11.27

      \[ \frac{\frac{\color{blue}{2 \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}}}{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \sqrt[3]{\tan k}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \sqrt[3]{\tan k}\right)\right)}^{2}} \]

      *-commutative [<=]11.27

      \[ \frac{\frac{\color{blue}{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot 2}}{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \sqrt[3]{\tan k}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \sqrt[3]{\tan k}\right)\right)}^{2}} \]

      associate-*l/ [=>]11.27

      \[ \frac{\frac{\color{blue}{\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot 2}{t}}}{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \sqrt[3]{\tan k}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \sqrt[3]{\tan k}\right)\right)}^{2}} \]

      associate-*r/ [<=]11.27

      \[ \frac{\frac{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \frac{2}{t}}}{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \sqrt[3]{\tan k}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \sqrt[3]{\tan k}\right)\right)}^{2}} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification12.38

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.6 \cdot 10^{+26}:\\ \;\;\;\;\frac{\frac{\frac{2}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \sqrt[3]{\tan k}}}{{\left(\frac{\sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}}}{\frac{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\sin k}}}\right)}^{2}}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-190}:\\ \;\;\;\;-2 \cdot \left(\frac{\frac{\cos k}{k}}{{\sin k}^{2}} \cdot \left(\ell \cdot \frac{\ell}{k \cdot \left(-t\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \frac{2}{t}}{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \sqrt[3]{\tan k}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \sqrt[3]{\tan k}\right)\right)}^{2}}\\ \end{array} \]

Alternatives

Alternative 1
Error12.39%
Cost98441
\[\begin{array}{l} t_1 := \sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \sqrt[3]{\tan k}\\ t_2 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ \mathbf{if}\;t \leq -5.4 \cdot 10^{+26} \lor \neg \left(t \leq 3.6 \cdot 10^{-190}\right):\\ \;\;\;\;\frac{\frac{t_2 \cdot \frac{2}{t}}{t_1}}{{\left(\frac{t}{t_2} \cdot t_1\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\frac{\frac{\cos k}{k}}{{\sin k}^{2}} \cdot \left(\ell \cdot \frac{\ell}{k \cdot \left(-t\right)}\right)\right)\\ \end{array} \]
Alternative 2
Error11.88%
Cost98252
\[\begin{array}{l} t_1 := \sqrt[3]{\tan k}\\ t_2 := \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\\ t_3 := {\sin k}^{2}\\ t_4 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_5 := \frac{\frac{2}{t_2}}{\sqrt[3]{t_4 \cdot \sin k} \cdot t_1}\\ t_6 := \frac{t_5}{{\left(\sqrt[3]{t_4 \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)}^{2}}\\ \mathbf{if}\;k \leq -2.1 \cdot 10^{+139}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\cos k}{k}}{k \cdot \left(t_3 \cdot \frac{t}{\ell}\right)}\\ \mathbf{elif}\;k \leq -5 \cdot 10^{-131}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;k \leq 1.65 \cdot 10^{-159}:\\ \;\;\;\;\frac{t_5}{{\left(t_2 \cdot \left(t_1 \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{2}\right)\right)\right)}^{2}}\\ \mathbf{elif}\;k \leq 3.8 \cdot 10^{+98}:\\ \;\;\;\;t_6\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot t_3}\right)\\ \end{array} \]
Alternative 3
Error13.22%
Cost92304
\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_3 := \frac{\frac{\frac{2}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{t_2 \cdot \sin k} \cdot \sqrt[3]{\tan k}}}{{\left(\sqrt[3]{t_2 \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)}^{2}}\\ \mathbf{if}\;k \leq -3 \cdot 10^{+134}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\cos k}{k}}{k \cdot \left(t_1 \cdot \frac{t}{\ell}\right)}\\ \mathbf{elif}\;k \leq -4.5 \cdot 10^{-146}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;k \leq 1.65 \cdot 10^{-159}:\\ \;\;\;\;\frac{\ell}{{\left(\frac{t}{\frac{\sqrt[3]{\ell}}{{\left(\sqrt[3]{k}\right)}^{2}}}\right)}^{3}}\\ \mathbf{elif}\;k \leq 2.2 \cdot 10^{+99}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot t_1}\right)\\ \end{array} \]
Alternative 4
Error13.32%
Cost85904
\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_3 := \frac{\frac{2}{t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \left(t_2 \cdot \tan k\right)}\right)}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{t_2 \cdot \left(\sin k \cdot \tan k\right)}\right)}^{2}}\\ \mathbf{if}\;k \leq -2 \cdot 10^{+143}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\cos k}{k}}{k \cdot \left(t_1 \cdot \frac{t}{\ell}\right)}\\ \mathbf{elif}\;k \leq -4.5 \cdot 10^{-146}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;k \leq 1.65 \cdot 10^{-159}:\\ \;\;\;\;\frac{\ell}{{\left(\frac{t}{\frac{\sqrt[3]{\ell}}{{\left(\sqrt[3]{k}\right)}^{2}}}\right)}^{3}}\\ \mathbf{elif}\;k \leq 4.8 \cdot 10^{+97}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot t_1}\right)\\ \end{array} \]
Alternative 5
Error13.42%
Cost46480
\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right) \cdot \tan k}\right)}^{3}}\\ \mathbf{if}\;k \leq -4.4 \cdot 10^{+138}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\cos k}{k}}{k \cdot \left(t_1 \cdot \frac{t}{\ell}\right)}\\ \mathbf{elif}\;k \leq -4.5 \cdot 10^{-146}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 1.5 \cdot 10^{-159}:\\ \;\;\;\;\frac{\ell}{{\left(\frac{t}{\frac{\sqrt[3]{\ell}}{{\left(\sqrt[3]{k}\right)}^{2}}}\right)}^{3}}\\ \mathbf{elif}\;k \leq 5.4 \cdot 10^{+97}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot t_1}\right)\\ \end{array} \]
Alternative 6
Error17.64%
Cost39948
\[\begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;k \leq -2.45 \cdot 10^{-20}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\cos k}{k}}{k \cdot \left(t_1 \cdot \frac{t}{\ell}\right)}\\ \mathbf{elif}\;k \leq 2.8 \cdot 10^{-96}:\\ \;\;\;\;\frac{\ell}{{\left({\left(\sqrt[3]{k}\right)}^{2} \cdot \frac{t}{\sqrt[3]{\ell}}\right)}^{3}}\\ \mathbf{elif}\;k \leq 1.55 \cdot 10^{+96}:\\ \;\;\;\;\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot t_1}\right)\\ \end{array} \]
Alternative 7
Error20.09%
Cost33680
\[\begin{array}{l} t_1 := {\left(\sqrt[3]{k}\right)}^{2}\\ t_2 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -6.8 \cdot 10^{+26}:\\ \;\;\;\;\frac{\ell}{{\left(t_1 \cdot \frac{t}{\sqrt[3]{\ell}}\right)}^{3}}\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-42}:\\ \;\;\;\;-2 \cdot \left(\frac{\frac{\cos k}{k}}{{\sin k}^{2}} \cdot \left(\ell \cdot \frac{\ell}{k \cdot \left(-t\right)}\right)\right)\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{+89}:\\ \;\;\;\;\frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\frac{\ell}{\frac{\sin k}{\ell}}}{t_2}\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{+205}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\sin k \cdot \left(t_2 \cdot \tan k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{\left(\frac{t}{\frac{\sqrt[3]{\ell}}{t_1}}\right)}^{3}}\\ \end{array} \]
Alternative 8
Error19.01%
Cost33672
\[\begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{+27}:\\ \;\;\;\;\frac{\ell}{{\left({\left(\sqrt[3]{k}\right)}^{2} \cdot \frac{t}{\sqrt[3]{\ell}}\right)}^{3}}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{-42}:\\ \;\;\;\;-2 \cdot \left(\frac{\frac{\cos k}{k}}{{\sin k}^{2}} \cdot \left(\ell \cdot \frac{\ell}{k \cdot \left(-t\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{t}^{-1.5}}{\tan k} \cdot \left(\frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{{t}^{-1.5} \cdot \left(2 \cdot \ell\right)}{\sin k}\right)\\ \end{array} \]
Alternative 9
Error18.97%
Cost27212
\[\begin{array}{l} t_1 := {\left(\sqrt[3]{k}\right)}^{2}\\ \mathbf{if}\;t \leq -3.3 \cdot 10^{+28}:\\ \;\;\;\;\frac{\ell}{{\left(t_1 \cdot \frac{t}{\sqrt[3]{\ell}}\right)}^{3}}\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-42}:\\ \;\;\;\;-2 \cdot \left(\frac{\frac{\cos k}{k}}{{\sin k}^{2}} \cdot \left(\ell \cdot \frac{\ell}{k \cdot \left(-t\right)}\right)\right)\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+105}:\\ \;\;\;\;\frac{\frac{\ell \cdot \left(2 \cdot {t}^{-3}\right)}{\tan k}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\sin k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{\left(\frac{t}{\frac{\sqrt[3]{\ell}}{t_1}}\right)}^{3}}\\ \end{array} \]
Alternative 10
Error20.07%
Cost26572
\[\begin{array}{l} t_1 := \frac{\ell}{{\left({\left(\sqrt[3]{k}\right)}^{2} \cdot \frac{t}{\sqrt[3]{\ell}}\right)}^{3}}\\ \mathbf{if}\;t \leq -6.8 \cdot 10^{+27}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-37}:\\ \;\;\;\;-2 \cdot \left(\frac{\frac{\cos k}{k}}{{\sin k}^{2}} \cdot \left(\ell \cdot \frac{\ell}{k \cdot \left(-t\right)}\right)\right)\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{+119}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)\right) \cdot \left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 11
Error20.06%
Cost26572
\[\begin{array}{l} t_1 := {\left(\sqrt[3]{k}\right)}^{2}\\ \mathbf{if}\;t \leq -2.1 \cdot 10^{+28}:\\ \;\;\;\;\frac{\ell}{{\left(t_1 \cdot \frac{t}{\sqrt[3]{\ell}}\right)}^{3}}\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{-37}:\\ \;\;\;\;-2 \cdot \left(\frac{\frac{\cos k}{k}}{{\sin k}^{2}} \cdot \left(\ell \cdot \frac{\ell}{k \cdot \left(-t\right)}\right)\right)\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{+119}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)\right) \cdot \left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{\left(\frac{t}{\frac{\sqrt[3]{\ell}}{t_1}}\right)}^{3}}\\ \end{array} \]
Alternative 12
Error22.62%
Cost21132
\[\begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{+27}:\\ \;\;\;\;\frac{\ell}{{\left(\sqrt[3]{\frac{k}{\ell}} \cdot \left(t \cdot \sqrt[3]{k}\right)\right)}^{3}}\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{-37}:\\ \;\;\;\;-2 \cdot \left(\frac{\frac{\cos k}{k}}{{\sin k}^{2}} \cdot \left(\ell \cdot \frac{\ell}{k \cdot \left(-t\right)}\right)\right)\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{+119}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)\right) \cdot \left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\frac{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}{\ell}}\\ \end{array} \]
Alternative 13
Error21.3%
Cost20620
\[\begin{array}{l} t_1 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{if}\;k \leq -4.2 \cdot 10^{+147}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -1.12 \cdot 10^{-20}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \cos k}{\frac{t}{\ell} \cdot {\left(k \cdot \sin k\right)}^{2}}\\ \mathbf{elif}\;k \leq 5.6 \cdot 10^{-8}:\\ \;\;\;\;\frac{\ell}{{\left(\sqrt[3]{\frac{k}{\ell}} \cdot \left(t \cdot \sqrt[3]{k}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 14
Error21.27%
Cost20620
\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot t_1}\right)\\ \mathbf{if}\;k \leq -5 \cdot 10^{+150}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -1.12 \cdot 10^{-20}:\\ \;\;\;\;2 \cdot \frac{\ell}{\frac{t}{\ell} \cdot \left(t_1 \cdot \left(k \cdot \frac{k}{\cos k}\right)\right)}\\ \mathbf{elif}\;k \leq 6 \cdot 10^{-9}:\\ \;\;\;\;\frac{\ell}{{\left(\sqrt[3]{\frac{k}{\ell}} \cdot \left(t \cdot \sqrt[3]{k}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 15
Error21.32%
Cost20620
\[\begin{array}{l} t_1 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ t_2 := {\sin k}^{2}\\ t_3 := t \cdot t_2\\ \mathbf{if}\;k \leq -4.5 \cdot 10^{+151}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t_3}{t_1}}\\ \mathbf{elif}\;k \leq -1.04 \cdot 10^{-20}:\\ \;\;\;\;2 \cdot \frac{\ell}{\frac{t}{\ell} \cdot \left(t_2 \cdot \left(k \cdot \frac{k}{\cos k}\right)\right)}\\ \mathbf{elif}\;k \leq 5.3 \cdot 10^{-8}:\\ \;\;\;\;\frac{\ell}{{\left(\sqrt[3]{\frac{k}{\ell}} \cdot \left(t \cdot \sqrt[3]{k}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{\cos k}{t_3}\right)\\ \end{array} \]
Alternative 16
Error21.34%
Cost20620
\[\begin{array}{l} t_1 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ t_2 := {\sin k}^{2}\\ t_3 := t \cdot t_2\\ \mathbf{if}\;k \leq -4 \cdot 10^{+147}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t_3}{t_1}}\\ \mathbf{elif}\;k \leq -7 \cdot 10^{-22}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \cos k}{t_2 \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot k\right)\right)}\\ \mathbf{elif}\;k \leq 5.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\ell}{{\left(\sqrt[3]{\frac{k}{\ell}} \cdot \left(t \cdot \sqrt[3]{k}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{\cos k}{t_3}\right)\\ \end{array} \]
Alternative 17
Error22.52%
Cost20552
\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := \frac{\cos k}{k}\\ \mathbf{if}\;k \leq -2.3 \cdot 10^{-21}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot t_2}{k \cdot \left(t_1 \cdot \frac{t}{\ell}\right)}\\ \mathbf{elif}\;k \leq 2.75 \cdot 10^{-8}:\\ \;\;\;\;\frac{\ell}{{\left(\sqrt[3]{\frac{k}{\ell}} \cdot \left(t \cdot \sqrt[3]{k}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\frac{t_2}{t_1} \cdot \left(\ell \cdot \frac{\ell}{k \cdot \left(-t\right)}\right)\right)\\ \end{array} \]
Alternative 18
Error23.43%
Cost20488
\[\begin{array}{l} \mathbf{if}\;t \leq -5.3 \cdot 10^{+29}:\\ \;\;\;\;\frac{\ell}{{\left(\sqrt[3]{\frac{k}{\ell}} \cdot \left(t \cdot \sqrt[3]{k}\right)\right)}^{3}}\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{+49}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\cos k}{k}}{k \cdot \left({\sin k}^{2} \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\frac{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}{\ell}}\\ \end{array} \]
Alternative 19
Error29.81%
Cost20360
\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot {t}^{1.5}}\\ \mathbf{if}\;t \leq -3.3 \cdot 10^{-34}:\\ \;\;\;\;\frac{\ell}{{\left(\sqrt[3]{\frac{k}{\ell}} \cdot \left(t \cdot \sqrt[3]{k}\right)\right)}^{3}}\\ \mathbf{elif}\;t \leq 7 \cdot 10^{+29}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{{\left(k \cdot \sin k\right)}^{2}} \cdot \left(\ell \cdot \frac{\ell}{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot t_1\\ \end{array} \]
Alternative 20
Error28.36%
Cost20360
\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot {t}^{1.5}}\\ \mathbf{if}\;t \leq -4.4 \cdot 10^{-37}:\\ \;\;\;\;\frac{\ell}{{\left(\sqrt[3]{\frac{k}{\ell}} \cdot \left(t \cdot \sqrt[3]{k}\right)\right)}^{3}}\\ \mathbf{elif}\;t \leq 3.15 \cdot 10^{+32}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \cos k}{\frac{t}{\ell} \cdot {\left(k \cdot \sin k\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot t_1\\ \end{array} \]
Alternative 21
Error32.1%
Cost19972
\[\begin{array}{l} t_1 := \cos \left(k + k\right)\\ t_2 := \frac{\cos k}{k \cdot k}\\ t_3 := \frac{\ell}{k \cdot {t}^{1.5}}\\ \mathbf{if}\;t \leq -2.2 \cdot 10^{-40}:\\ \;\;\;\;\frac{\ell}{{\left(\sqrt[3]{\frac{k}{\ell}} \cdot \left(t \cdot \sqrt[3]{k}\right)\right)}^{3}}\\ \mathbf{elif}\;t \leq 1.12 \cdot 10^{-108}:\\ \;\;\;\;2 \cdot \left(t_2 \cdot \frac{2 \cdot \left(\ell \cdot \frac{\ell}{t}\right)}{1 - t_1}\right)\\ \mathbf{elif}\;t \leq 580:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right) \cdot \frac{k}{\frac{\ell}{t} \cdot \frac{\ell}{t \cdot t}}}\\ \mathbf{elif}\;t \leq 7 \cdot 10^{+29}:\\ \;\;\;\;2 \cdot \left(t_2 \cdot \frac{\frac{\ell}{\frac{t}{\ell}}}{0.5 - \frac{t_1}{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot t_3\\ \end{array} \]
Alternative 22
Error32.59%
Cost19908
\[\begin{array}{l} t_1 := \frac{\cos k}{k \cdot k}\\ t_2 := \frac{\ell}{k \cdot {t}^{1.5}}\\ t_3 := \cos \left(k + k\right)\\ \mathbf{if}\;t \leq -1.45 \cdot 10^{-30}:\\ \;\;\;\;\frac{\ell}{\frac{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}{\ell}}\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{-107}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{2 \cdot \left(\ell \cdot \frac{\ell}{t}\right)}{1 - t_3}\right)\\ \mathbf{elif}\;t \leq 650:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right) \cdot \frac{k}{\frac{\ell}{t} \cdot \frac{\ell}{t \cdot t}}}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{+29}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{\frac{\ell}{\frac{t}{\ell}}}{0.5 - \frac{t_3}{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot t_2\\ \end{array} \]
Alternative 23
Error34.23%
Cost14732
\[\begin{array}{l} t_1 := \frac{\cos k}{k \cdot k}\\ t_2 := \frac{\ell}{k \cdot {t}^{1.5}}\\ t_3 := \cos \left(k + k\right)\\ \mathbf{if}\;t \leq -7 \cdot 10^{-44}:\\ \;\;\;\;\frac{\ell}{k \cdot \frac{k \cdot {t}^{3}}{\ell}}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-108}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{2 \cdot \left(\ell \cdot \frac{\ell}{t}\right)}{1 - t_3}\right)\\ \mathbf{elif}\;t \leq 350:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right) \cdot \frac{k}{\frac{\ell}{t} \cdot \frac{\ell}{t \cdot t}}}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{+29}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{\frac{\ell}{\frac{t}{\ell}}}{0.5 - \frac{t_3}{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot t_2\\ \end{array} \]
Alternative 24
Error33.62%
Cost14672
\[\begin{array}{l} t_1 := 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{2 \cdot \left(\ell \cdot \frac{\ell}{t}\right)}{1 - \cos \left(k + k\right)}\right)\\ t_2 := \frac{\ell}{k \cdot {t}^{1.5}}\\ \mathbf{if}\;t \leq -1.4 \cdot 10^{-43}:\\ \;\;\;\;\frac{\ell}{k \cdot \frac{k \cdot {t}^{3}}{\ell}}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-107}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{+17}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{+30}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot t_2\\ \end{array} \]
Alternative 25
Error33.63%
Cost14672
\[\begin{array}{l} t_1 := \frac{\cos k}{k \cdot k}\\ t_2 := \frac{\ell}{k \cdot {t}^{1.5}}\\ t_3 := \cos \left(k + k\right)\\ \mathbf{if}\;t \leq -4.3 \cdot 10^{-45}:\\ \;\;\;\;\frac{\ell}{k \cdot \frac{k \cdot {t}^{3}}{\ell}}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-107}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{2 \cdot \left(\ell \cdot \frac{\ell}{t}\right)}{1 - t_3}\right)\\ \mathbf{elif}\;t \leq 5.3 \cdot 10^{+17}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{+29}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{\frac{\ell}{\frac{t}{\ell}}}{0.5 - \frac{t_3}{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot t_2\\ \end{array} \]
Alternative 26
Error34.28%
Cost13896
\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot {t}^{1.5}}\\ \mathbf{if}\;t \leq -2.7 \cdot 10^{-40}:\\ \;\;\;\;\frac{\ell}{k \cdot \frac{k \cdot {t}^{3}}{\ell}}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-39}:\\ \;\;\;\;-2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\left(\ell \cdot \frac{\ell}{t}\right) \cdot -0.3333333333333333 - \frac{\ell}{k} \cdot \frac{\frac{\ell}{t}}{k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot t_1\\ \end{array} \]
Alternative 27
Error34.88%
Cost13512
\[\begin{array}{l} \mathbf{if}\;t \leq -4.1 \cdot 10^{-32}:\\ \;\;\;\;\frac{\ell}{k \cdot \frac{k \cdot {t}^{3}}{\ell}}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-38}:\\ \;\;\;\;-2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\left(\ell \cdot \frac{\ell}{t}\right) \cdot -0.3333333333333333 - \frac{\ell}{k} \cdot \frac{\frac{\ell}{t}}{k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{\ell}{k}}{{t}^{1.5}}\right)}^{2}\\ \end{array} \]
Alternative 28
Error36.99%
Cost8265
\[\begin{array}{l} \mathbf{if}\;t \leq -2.35 \cdot 10^{-40} \lor \neg \left(t \leq 1.36 \cdot 10^{-30}\right):\\ \;\;\;\;\frac{\ell}{k \cdot \frac{k \cdot {t}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\left(\ell \cdot \frac{\ell}{t}\right) \cdot 0.3333333333333333 + \frac{\ell}{t} \cdot \frac{\frac{\ell}{k}}{k}\right)\right)\\ \end{array} \]
Alternative 29
Error36.98%
Cost8265
\[\begin{array}{l} \mathbf{if}\;t \leq -6.8 \cdot 10^{-38} \lor \neg \left(t \leq 3.9 \cdot 10^{-30}\right):\\ \;\;\;\;\frac{\ell}{k \cdot \frac{k \cdot {t}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\left(\ell \cdot \frac{\ell}{t}\right) \cdot -0.3333333333333333 - \frac{\ell}{k} \cdot \frac{\frac{\ell}{t}}{k}\right)\right)\\ \end{array} \]
Alternative 30
Error38.25%
Cost8137
\[\begin{array}{l} \mathbf{if}\;t \leq -8.5 \cdot 10^{-34} \lor \neg \left(t \leq 3.8 \cdot 10^{-28}\right):\\ \;\;\;\;\frac{\ell}{k \cdot \frac{k \cdot {t}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k} + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)}{t}\right)\\ \end{array} \]
Alternative 31
Error38.24%
Cost7753
\[\begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{-41} \lor \neg \left(t \leq 1.7 \cdot 10^{-30}\right):\\ \;\;\;\;\frac{\ell}{k \cdot \frac{k \cdot {t}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\frac{\ell}{\frac{t}{\ell}}}{k \cdot k}\right)\\ \end{array} \]
Alternative 32
Error40.41%
Cost7305
\[\begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{-47} \lor \neg \left(t \leq 2.75 \cdot 10^{-30}\right):\\ \;\;\;\;\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot \frac{k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \end{array} \]
Alternative 33
Error38.13%
Cost7305
\[\begin{array}{l} \mathbf{if}\;t \leq -1.35 \cdot 10^{-43} \lor \neg \left(t \leq 2.6 \cdot 10^{-38}\right):\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \end{array} \]
Alternative 34
Error38.5%
Cost7305
\[\begin{array}{l} \mathbf{if}\;t \leq -2.95 \cdot 10^{-34} \lor \neg \left(t \leq 2.3 \cdot 10^{-29}\right):\\ \;\;\;\;\frac{\ell}{k \cdot \frac{k \cdot {t}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \end{array} \]
Alternative 35
Error38.29%
Cost7304
\[\begin{array}{l} t_1 := k \cdot {t}^{3}\\ \mathbf{if}\;t \leq -6 \cdot 10^{-44}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k}}{t_1}\\ \mathbf{elif}\;t \leq 1.32 \cdot 10^{-37}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{t_1}\\ \end{array} \]
Alternative 36
Error41.84%
Cost1865
\[\begin{array}{l} \mathbf{if}\;t \leq -1.3 \cdot 10^{-40} \lor \neg \left(t \leq 1.8 \cdot 10^{-30}\right):\\ \;\;\;\;\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot \frac{k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{1}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{t \cdot \left(k \cdot k\right)} - \left(\ell \cdot \frac{\ell}{t}\right) \cdot -0.3333333333333333\right)\right)\\ \end{array} \]
Alternative 37
Error46.82%
Cost1097
\[\begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-168} \lor \neg \left(\ell \leq 2.12 \cdot 10^{-162}\right):\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;-0.11666666666666667 \cdot \frac{\ell \cdot \ell}{t}\\ \end{array} \]
Alternative 38
Error42.9%
Cost1097
\[\begin{array}{l} \mathbf{if}\;k \leq -5.2 \lor \neg \left(k \leq 2.6 \cdot 10^{-129}\right):\\ \;\;\;\;\frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot \frac{k}{\ell}\right)\right)}\\ \end{array} \]
Alternative 39
Error47.1%
Cost1096
\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.05 \cdot 10^{-168}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t \cdot t}\\ \mathbf{elif}\;\ell \leq 1.65 \cdot 10^{-162}:\\ \;\;\;\;-0.11666666666666667 \cdot \frac{\ell \cdot \ell}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t \cdot t} \cdot \frac{\ell}{t \cdot \left(k \cdot k\right)}\\ \end{array} \]
Alternative 40
Error47.01%
Cost1096
\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.05 \cdot 10^{-168}:\\ \;\;\;\;\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot \frac{k}{\ell}\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.85 \cdot 10^{-162}:\\ \;\;\;\;-0.11666666666666667 \cdot \frac{\ell \cdot \ell}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t \cdot t} \cdot \frac{\ell}{t \cdot \left(k \cdot k\right)}\\ \end{array} \]
Alternative 41
Error66.96%
Cost576
\[\frac{1}{\frac{t}{\left(\ell \cdot \ell\right) \cdot -0.11666666666666667}} \]
Alternative 42
Error70.15%
Cost448
\[\left(\ell \cdot \frac{\ell}{t}\right) \cdot -0.11666666666666667 \]
Alternative 43
Error67.23%
Cost448
\[-0.11666666666666667 \cdot \frac{\ell \cdot \ell}{t} \]
Alternative 44
Error67%
Cost448
\[\frac{-0.11666666666666667}{\frac{t}{\ell \cdot \ell}} \]

Error

Reproduce?

herbie shell --seed 2023090 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  :precision binary64
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))